3D Cartesian Vectors Overview
26 Questions
0 Views

3D Cartesian Vectors Overview

Created by
@ValuableHeliotrope5203

Podcast Beta

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the correct Cartesian vector form for F1?

  • 0i + 500j + 200k lb
  • 0i + 300j + 400k lb
  • 0i + 200j + 500k lb
  • 0i + 400j + 300k lb (correct)

    • For force F2, what is the value of F2z resolved into the z-axis?

  • 489.90 lb
  • 565.69 lb (correct)
  • 282.84 lb
  • 800 lb

    • What is the result of summing F1 and F2 in components?

  • 491i + 683j + 300k lb
  • 490i + 566j - 300k lb
  • 0i + 683j + 266k lb
  • 490i + 683j - 266k lb (correct)

    • How is F2 resolved into its components in the xy-plane?

    <p>F2x = F2 * cos 30 and F2y = F2 * sin 30</p> Signup and view all the answers

    What components of F1 are responsible for the j-direction?

    <p>400 lb</p> Signup and view all the answers

    What is the magnitude of the x-component of F2?

    <p>489.90 lb</p> Signup and view all the answers

    What does the resultant force vector FR likened to in this problem represent?

    <p>The net force acting on the system</p> Signup and view all the answers

    Which direction does the k-component of the resultant force FR act in?

    <p>Negative direction</p> Signup and view all the answers

    What is the formula to express the position vector from point A to point B?

    <p>$\mathbf{r_{AB}} = (x_B - x_A) \mathbf{i} + (y_B - y_A) \mathbf{j} + (z_B - z_A) \mathbf{k}$</p> Signup and view all the answers

    How is the magnitude of the position vector $\mathbf{r_{AB}}$ calculated?

    <p>$r_{AB} = \sqrt{(x_B - x_A) ^ 2 + (y_B - y_A) ^ 2 + (z_B - z_A) ^ 2}$</p> Signup and view all the answers

    What represents the direction of the position vector $\mathbf{r_{AB}}$?

    <p>$\mathbf{u} = \frac{\mathbf{r_{AB}}}{r_{AB}}$</p> Signup and view all the answers

    If force $\mathbf{F}$ acts along the line from point A to point B, how can it be expressed using position vector?

    <p>$\mathbf{F} = F \mathbf{u}$</p> Signup and view all the answers

    Which statement accurately describes how to find the distance between two points A and B?

    <p>You should calculate the magnitude of the position vector directed from A to B.</p> Signup and view all the answers

    What is the role of position vectors in three-dimensional statics concerning force representation?

    <p>They represent the line of action of forces passing through points A and B.</p> Signup and view all the answers

    Which of the following is NOT part of the formula to compute the direction cosines of the position vector $\mathbf{r_{AB}}$?

    <p>$\sin \theta$</p> Signup and view all the answers

    When determining the resultant force at point A, which of the following calculations is necessary?

    <p>Finding the vector addition of forces at points A and B.</p> Signup and view all the answers

    What is the correct representation of a 3D Cartesian vector A?

    <p>𝐀 = 𝐴𝑥 𝐢 + 𝐴𝑦 𝐣 + 𝐴𝑧 𝐤</p> Signup and view all the answers

    Which method is used to calculate the magnitude of a Cartesian vector?

    <p>Pythagorean Theorem</p> Signup and view all the answers

    What are the coordinate direction angles of a Cartesian vector A represented by?

    <p>Angles with respect to the x, y, and z axes</p> Signup and view all the answers

    Which expression correctly represents the unit vector uA in the direction of vector A?

    <p>𝐮𝐀 = cos 𝛼𝐢 + cos 𝛽𝐣 + cos 𝛾𝐤</p> Signup and view all the answers

    How can the resultant vector R of two vectors A and B be found?

    <p>By summing their components: 𝐑 = 𝐀 + 𝐁</p> Signup and view all the answers

    What is the relationship between the angles α, β, and γ in terms of a unit vector?

    <p>cos² α + cos² β + cos² γ = 1</p> Signup and view all the answers

    In spherical vector representation, what determines the x and y components?

    <p>The angles θ and φ</p> Signup and view all the answers

    What does the right-handed coordinate system indicate about the orientation of the axes?

    <p>Fingers curl from the x-axis towards the y-axis</p> Signup and view all the answers

    How would you express the addition of vectors A and B in Cartesian components?

    <p>𝐑 = 𝐀 + 𝐁 = (𝐴𝑥 + 𝐵𝑥 )𝐢 + (𝐴𝑦 + 𝐵𝑦 )𝐣 + (𝐴𝑧 + 𝐵𝑧 )𝐤</p> Signup and view all the answers

    Which of the following statements is true regarding coordinate directional angles?

    <p>Each angle can only be calculated if the other two are known.</p> Signup and view all the answers

    Study Notes

    3D Cartesian Vectors

    • 3D vectors are best represented in Cartesian notation.
    • A right-handed coordinate system is used.
      • Thumb represents the positive z-axis.
      • Fingers curl around the z-axis.
      • Sweep fingers from the x-axis to the y-axis.
    • Cartesian Unit Vectors i, j, k designate x, y, and z directions.
      • The positive directions of the unit vectors are shown in the provided figure.

    Cartesian Vector Representation

    • Resolve a vector A into Cartesian unit vectors using the parallelogram law.
    • The equation for vector A is 𝐀 = 𝐴 𝑥 𝐢 + 𝐴 𝑦 𝐣 + 𝐴𝑧 𝐤.
    • The magnitude of a Cartesian vector is determined using the Pythagorean Theorem.
      • 𝐀 = 𝐴2𝑥 + 𝐴2𝑦 + 𝐴𝑧2
    • The direction of a Cartesian vector is defined by angles , , and  with respect to the x, y, and z axes, called coordinate direction angles.
      • cos 𝛼 = 𝐴𝑥 / 𝐴
      • cos 𝛽 = 𝐴𝑦 / 𝐴
      • cos 𝛾 = 𝐴𝑧 / 𝐴
    • A vector A can be represented using unit vectors as:
      • 𝐀 = 𝐴𝐮𝐀
      • where uA is a unit vector in the direction of A
      • 𝐮𝐀 = 𝐴 / 𝐴 = 𝐴𝑥 / 𝐴 𝐢 + 𝐴𝑦 / 𝐴 𝐣 + 𝐴𝑧 / 𝐴 𝐤
      • 𝐮𝐀 = cos 𝛼𝐢 + cos 𝛽𝐣 + cos 𝛾𝐤.
    • The magnitude of a unit vector is 1.
    • If two coordinate angles are known, the third can be found using the equation:
      • cos2 𝛼 + cos2 𝛽 + cos2 𝛾 = 1

    Spherical Vector Representation

    • The direction of a vector A can be found from angles  and .
      • 𝐴𝑧 = 𝐀 cos 𝜙
      • 𝐴′ = 𝐀 sin 𝜙
      • 𝐴𝑥 = 𝐀′ cos 𝜃 = 𝐀 sin 𝜙 cos 𝜃
      • 𝐴𝑦 = 𝐀′ sin 𝜃 = 𝐀 sin 𝜙 sin 𝜃
      • 𝐀 = 𝐴 sin 𝜙 cos 𝜃 𝐢 + 𝐴 sin 𝜙 sin 𝜃 𝐣 + A 𝑐𝑜𝑠𝜙 𝐤

    Cartesian Vector Addition

    • Given two vectors A and B:
      • 𝐀 = 𝐴𝑥 𝐢 + 𝐴𝑦 𝐣 + 𝐴𝑧 𝐤
      • 𝐁 = 𝐵𝑥 𝐢 + 𝐵𝑦 𝐣 + 𝐵𝑧 𝐤.
    • Add A and B using Cartesian components:
      • 𝐑 = 𝐀 + 𝐁 = (𝐴𝑥 +𝐵𝑥 )𝐢 + (𝐴𝑦 +𝐵𝑦 )𝐣 + (𝐴𝑧 +𝐵𝑧 )𝐤
    • Subtract A and B using Cartesian components:
      • 𝐑 = 𝐀 − 𝐁 = (𝐴𝑥 −𝐵𝑥 )𝐢 + (𝐴𝑦 −𝐵𝑦 )𝐣 + (𝐴𝑧 −𝐵𝑧 )𝐤
    • 𝐑 = ∑𝐅 = ∑𝐹𝑥 𝐢 + ∑𝐹𝑦 𝐣 + ∑𝐹𝑧 𝐤

    Position Vectors

    • A position vector r is a fixed vector which defines a point in 3D space relative to another point.
      • For example, a point P relative to the origin O.
      • O (0, 0, 0) and P (x, y, z)
      • 𝐫 = 𝑥𝐢 + 𝑦𝐣 + 𝑧𝐤
    • If a position vector is directed from point A (xA, yA, zA) to B (xB, yB, zB), then:
      • 𝐫𝐀𝐁 = 𝐫𝐁 − 𝐫𝐀
      • 𝐫𝐀𝐁 = 𝑥𝐵 𝐢 + 𝑦𝐵 𝐣 + 𝑧𝐵 𝐤 − 𝑥𝐴 𝐢 + 𝑦𝐴 𝐣 + 𝑧𝐴 𝐤
      • 𝐫𝐀𝐁 = 𝑥𝐵 − 𝑥𝐴 ) 𝐢 + (𝑦𝐵 − 𝑦𝐴 )𝐣 + (𝑧𝐵 −𝑧𝐴 𝐤
    • Position vectors tell you how to get from A to B.
    • The magnitude of the position vector rAB is:
      • 𝑟𝐴𝐵 = 𝑥𝐵 − 𝑥𝐴 2 + 𝑦𝐵 − 𝑦𝐴 2 + 𝑧𝐵 − 𝑧𝐴 2
    • The direction of the position vector rAB is described by the direction cosines of rAB, as specified by the unit vector:
      • 𝐮= 𝐫𝐀𝐁 / 𝑟𝐴𝐵 = cos 𝛼𝐢 + cos 𝛽𝐣 + cos 𝛾𝐤

    Force Along a Line

    • In 3-dimensional statics, the force F can be specified by two points, A and B, through which passes the line of action of F.
    • F can be represented by the position vector r from point A to B.
    • 𝐅 = 𝐹𝐮 = 𝐹 𝐫 / 𝑟
    • 𝐅 = 𝐹 (𝑥𝐵 − 𝑥𝐴 ) 𝐢 + (𝑦𝐵 − 𝑦𝐴 )𝐣 + (𝑧𝐵 −𝑧𝐴 𝐤 / 𝑥𝐵 − 𝑥𝐴 2 + 𝑦𝐵 − 𝑦𝐴 2 + 𝑧𝐵 − 𝑧𝐴 2

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    ECOR 1045 Lecture 3 PDF

    Description

    Explore the fundamentals of 3D Cartesian vectors in this quiz. Learn how to represent vectors using Cartesian notation, resolve them into unit vectors, and compute their magnitudes and directions. This quiz will enhance your understanding of vector representation in three-dimensional space.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser